INMO 2025

Consider the sequence defined by $a_1 = 2$, $a_2 = 3$, and $$a_{2k+1} = 2 + 2a_k, a_{2k+2} = 2 + a_k + a_{k+1},$$ for all integers $k \geq 1$. Determine all positive integers $n$ such that $$\frac{a_n}{n}$$ is an integer.

Proposed by Niranjan Balachandran, SS Krishnan, and Prithwijit De.

Let $n\ge 2$ be a positive integer. The integers $1,2,\cdots,n$ are written on a board. In a move, Alice can pick two integers written on the board $a\neq b$ such that $a+b$ is an even number, erase both $a$ and $b$ from the board and write the number $\frac{a+b}{2}$ on the board instead. Find all $n$ for which Alice can make a sequence of moves so that she ends up with only one number remaining on the board.Note. When $n=3$ , Alice changes $(1,2,3)$ to $(2,2)$ and can't make any further moves.

Proposed by Rohan Goyal

Euclid has a tool called splitter which can only do the following two types of operations : • Given three non-collinear marked points $X,Y,Z$ it can draw the line which forms the interior angle bisector of $\angle{XYZ}$ . • It can mark the intersection point of two previously drawn non-parallel lines . Suppose Euclid is only given three non-collinear marked points $A,B,C$ in the plane . Prove that Euclid can use the splitter several times to draw the centre of circle passing through $A,B$ and $C$ .

Proposed by Shankhadeep Ghosh

Let $n\ge 3$ be a positive integer. Find the largest real number $t_n$ as a function of $n$ such that the inequality $$\max\left(|a_1+a_2|, |a_2+a_3|, \dots ,|a_{n-1}+a_{n}| , |a_n+a_1|\right) \ge t_n \cdot \max(|a_1|,|a_2|, \dots ,|a_n|)$$ holds for all real numbers $a_1, a_2, \dots , a_n$ .

Proposed by Rohan Goyal and Rijul Saini

Greedy goblin Griphook has a regular $2000$ -gon, whose every vertex has a single coin. In a move, he chooses a vertex, removes one coin each from the two adjacent vertices, and adds one coin to the chosen vertex, keeping the remaining coin for himself. He can only make such a move if both adjacent vertices have at least one coin. Griphook stops only when he cannot make any more moves. What is the maximum and minimum number of coins he could have collected?

Proposed by Pranjal Srivastava and Rohan Goyal

Let $b \geqslant 2$ be a positive integer. Anu has an infinite collection of notes with exactly $b-1$ copies of a note worth $b^k-1$ rupees, for every integer $k\geqslant 1$ . A positive integer $n$ is called payable if Anu can pay exactly $n^2+1$ rupees by using some collection of her notes. Prove that if there is a payable number, there are infinitely many payable numbers.

Proposed by Shantanu Nene